A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems

Abstract

AbstractLetEbe a uniformly convex and uniformly smooth real Banach space, and letE* be its dual. LetA : E→ 2E*be a bounded maximal monotone map. Assume thatA−1(0) ≠ Ø. A new iterative sequence is constructed which convergesstronglyto an element ofA−1(0). The theorem proved complements results obtained on strong convergence ofthe proximal point algorithmfor approximating an element ofA−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.